Minimal simplicial spherical mappings with a given degree

2025/11/14 by Apolonskaya, Ksenia, Musin, Oleg R. · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.10870

Abstract

This paper studies the minimal number of vertices λ(n,d) required in a triangulation of the n-sphere to admit a simplicial map to the boundary of a (n+1)-simplex with a given degree d. We establish upper bounds for λ(n,d) in dimensions n ≥ 3. Furthermore, we provide exact formulas for small values of d, showing that λ(n,d)=n+d+3 for n ≥ 3 and d=2,3,4. A key technical result is the identity λ(n,d) = λ(d-1,d) + n - d + 1 for n ≥ d, which allows us to reduce higher-dimensional cases to lower-dimensional ones. The proofs involve constructive methods based on local modifications of triangulations and combinatorial arguments.

Citations

• On vertex-minimal simplicial maps to the sphere
• Simplicial degree d self-maps on n-spheres
• Homotopy groups and quantitative Sperner-type lemma
• Around Sperner's lemma
• Minimal Simplicial Self-maps of the 2-Sphere

Cited by

• Computing the Hopf invariant
• On vertex-minimal simplicial maps to the sphere

Related

• On vertex-minimal simplicial maps to the sphere
• Simplicial degree d self-maps on n-spheres
• Minimal triangulations of sphere bundles over the circle
• Simplicial maps between spheres and Davis' manifolds with positive simplicial volume
• Vertex-Minimal Triangulation of Complexes with Homology